Negative Sampling and Probability Recalibration

A CTR model is trained on 2 billion impressions per day with a 0.5 % click rate. To keep training tractable, the team keeps all positives and 5 % of negatives. The trained model outputs p' = 0.40 for a given impression.

1. What is the effective positive rate in the training set, and what is the corrected probability p for that impression?
2. Why does the correction matter for this system, and when could you skip it?
3. The same team wants to build a two-tower retrieval model over a catalogue of 10 M items. Explain why "impressed-not-clicked" rows alone are not sufficient negatives, and propose a negative mix.